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Optimal Stopping Problems for Time-Homogeneous Diffusions: A Review

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Recent Advances in Applied Probability

Abstract

The first part of this paper summarizes the essential facts on general optimal stopping theory for time-homogeneous diffusion processes in ℝn. The results displayed are stated in a little greater generality, but in such a way that they are neither too restrictive nor too complicated. The second part presents equations for the value function and the optimal stopping boundary as a free-boundary (Stefan) problem and further presents the principle of smooth fit. This part is illustrated by examples where the focus is on optimal stopping problems for the maximum process associated with a one-dimensional diffusion.

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References

  • Chernoff, H. (1961). Sequential tests for the mean of a normal distribution. Proc. 4th Berkeley Sympos. Math. Statist. and Prob.1, Univ. California Press (79–91).

    MATH  MathSciNet  Google Scholar 

  • Conze, A.andViswanathan, R. (1991). Path dependent options: The case of lookback options. J. Finance46 (1893–1907).

    Google Scholar 

  • Davis, B. (1976). On the Lp norms of stochastic integrals and other martingales. Duke Math. J.43 (697–704).

    Article  MATH  MathSciNet  Google Scholar 

  • Dubins, L.E., Shepp, L.A.andShiryaev, A.N. (1993). Optimal stopping rules and maximal inequalities for Bessel processes. Theory Probab. Appl.38 (226–261).

    Article  MathSciNet  Google Scholar 

  • Dynkin, E.B. (1963). Optimal choice of the stopping moment of a Markov process. Dokl. Akad. Nauk SSSR150 (238–240). (In Russian).

    MATH  MathSciNet  Google Scholar 

  • El Karoui, N. (1981). Les aspects probabilites du contrôle stochastique. Lecture Notes in Math.876, Springer (73–238). (In French).

    Google Scholar 

  • Graversen, S.E.andPeskir, G. (1997). On Doob’s maximal inequality for Brownian motion. Stochastic Process. Appl.69 (111–125).

    Article  MathSciNet  Google Scholar 

  • Graversen, S.E.andPeskir, G. (1998). Optimal stopping and maximal inequalities for geometric Brownian motion. J. Appl. Probab.42 (564–575).

    Google Scholar 

  • Graversen, S.E.andPeskir, G. (1998a). Optimal stopping in the L log L-inequality of Hardy and Littlewood. Bull. London Math. Soc.30 (171–181).

    Article  MathSciNet  Google Scholar 

  • Graversen, S.E., Peskir, G.andShiryaev, A.N. (2001). Stopping Brownian motion without anti-cipation as close as possible to its ultimate maximum. Theory Probab. Appl.45 (41–50).

    Article  MathSciNet  Google Scholar 

  • Grigelionis, B.I.andShiryaev, A.N. (1966). On the Stefan problem and optimal stopping rules for Markov processes. Theory Probab. Appl.11 (541–558).

    Article  Google Scholar 

  • Jacka, S.D. (1991). Optimal stopping and bests constant for Doob-like inequalities I: The case p= 1.Ann.Probab.19 (1798–1821).

    MATH  MathSciNet  Google Scholar 

  • Lindley, D.V. (1961). Dynamic programming and decision theory. Appl. Statist.10 (39–51).

    MATH  MathSciNet  Google Scholar 

  • McKean, H.P. (1965). A free-boundary problem for the heat equation arising from the problem of mathematical economics. Industrial Managem. Review6 (32–39).

    MathSciNet  Google Scholar 

  • Mikhalevich, V.S. (1958). Bayesian choice between two hypotheses for the mean value of a normal process. Visnik Kiiv. Univ.1 (101–104). (in Ukrainian).

    Google Scholar 

  • Myneni, R. (1992). The pricing of the American option. Ann. Appl. Probab.2 (1–23).

    MATH  MathSciNet  Google Scholar 

  • Ksendal, B. (1998). Stochastic differential equations. An introduction with applications. (Fifth edition). Springer.

    Google Scholar 

  • Pedersen, J.L. (2000). Best bounds in Doob’s maximal inequality for Bessel processes. J. Multivariate Anal.75 (36–46).

    Article  MATH  MathSciNet  Google Scholar 

  • Pedersen, J.L. (2000a). Discounted optimal stopping problems for the maximum process. J. Appl. Probab.37 (972–983).

    MATH  MathSciNet  Google Scholar 

  • Pedersen, J.L. (2003). Optimal prediction of the ultimate maximum of Brownian motion. Stoch. Stoch Rep.75 (205–219).

    MATH  MathSciNet  Google Scholar 

  • Pedersen, J.L.andPeskir, G. (1998). Computing the expectation of the Azéma-Yor stopping time. Ann. Inst. H. Poincare Probab. Statist.34 (265–276).

    MathSciNet  Google Scholar 

  • Pedersen, J.L.andPeskir, G. (2000). Solving non-linear optimal stopping problems by the method of time-change. Stochastic Anal. Appl.18 (811–835).

    MathSciNet  Google Scholar 

  • Peskir, G. (1998). Optimal stopping of the maximum process: The maximality principle. Ann. Probab.26 (1614–1640).

    Article  MATH  MathSciNet  Google Scholar 

  • Peskir, G.andShiryaev, A.N. (2000). Sequential testing problems for Poisson processes. Ann. Statist.28 (837–859).

    MathSciNet  Google Scholar 

  • Shepp, L.A. (1969). Explicit solutions to some problems of optimal stopping. Ann. Math. Statist.40 (993–1010).

    MATH  MathSciNet  Google Scholar 

  • Shepp, L.A.andShiryaev, A.N. (1993). The Russian option: Reduced regret. Ann. Appl. Probab.3 (631–640).

    MathSciNet  Google Scholar 

  • Shiryaev, A.N. (1961). The problem of quickest detection of a violation of stationary behavior. Dokl. Akad. Nauk SSSR138 (1039–1042). (In Russian).

    MATH  MathSciNet  Google Scholar 

  • Shiryaev, A.N. (1969). Two problems of sequential analysis. Cybernetics3 (63–69).

    Article  MathSciNet  Google Scholar 

  • Shiryaev, A.N. (1978). Optimal stopping rules. Springer.

    Google Scholar 

  • Snell, J.L. (1952). Application of martingale system theorems. Trans. Amer. Math. Soc.73 (293–312).

    MATH  MathSciNet  Google Scholar 

  • Taylor, H.M. (1968). Optimal stopping in a Markov process. Ann. Math. Statist.39 (1333–1344).

    MATH  MathSciNet  Google Scholar 

  • Van Moerbeke, P. (1974). Optimal stopping and free boundary problems. Rocky Mountain. J. Math.4 (539–578).

    MATH  MathSciNet  Google Scholar 

  • Wald, A. (1947). Sequential Analysis. John Wiley and Sons.

    Google Scholar 

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Authors and Affiliations

  1. RiskLab, Department of Mathematics, ETH-Zentrum, CH-8092, Zürich, Switzerland

    Jesper Lund Pedersen

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  1. Jesper Lund Pedersen
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Editor information

Editors and Affiliations

  1. Universidad de Chile, Chile

    Ricardo Baeza-Yates

  2. University of Connecticut, USA

    Joseph Glaz

  3. Universidad Simón Bolívar, Venezuela

    Henryk Gzyl  & José Luis Palacios  & 

  4. University of Bern, Switzerland

    Jürgen Hüsler

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© 2005 Springer Science + Business Media, Inc.

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Pedersen, J.L. (2005). Optimal Stopping Problems for Time-Homogeneous Diffusions: A Review. In: Baeza-Yates, R., Glaz, J., Gzyl, H., Hüsler, J., Palacios, J.L. (eds) Recent Advances in Applied Probability. Springer, Boston, MA. https://doi.org/10.1007/0-387-23394-6_18

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